Consider a tube which is 3 ft wide and 18 ft long.

Find the surface area of the largest sphere which could fit within the tube described above.

Okay, so we need to find the surface area of a sphere. To do so, we need the following formula:

Now, you're probably thinking, "How do we find our radius?" Well, we need to look at our tube.

The largest sphere which will fit within the tube is the same thing as a sphere with diameter equal to the tube's diameter.

In this case, the diameter of the tube is 3 ft. This means our sphere's diameter is also 3 feet.

If our diameter is 3 ft, then our radius is half of that- 1.5 ft.

So, with a 1.5 ft radius, our surface area becomes:

So, leaving our answer in terms of pi, we get:

Find the surface area of a rectangular prism with the following dimensions: 6 ft by 12 ft by 4 ft.

To find the surface area of a rectangular prism, we essentially need to add up the area of all the sides. To do so, use the following formula:

Where l, w, and h are our length, height, and width. 

We simply need to plug in our given measurements and solve

So, our surface area is 

The point can be reached from the origin by moving to the right 7 units, making 7 the -coordinate. All points on the -axis, such as this one, have -coordinate 0. That makes this point's coordinates .

The point can be reached from the origin by moving left 7 units, making the first coordinate nagative 7, and up 3 units, making the second coordinate positive 3. The correct coordinates are .

The point  can be reached from the origin  by moving negative  2 units horizontally - that is, 2 to the left - then negative 4 units vertically - that is, 4 units down. This is point .

The point  can be reached from the origin  by moving 4 units horizontally in a positive direction - that is, 4 units to the right - then 2 units vertically in a positive direction - that is, 2 units up. This is point .

Recall that at a point of intersection of two lines, they will have the same x and y-coordinates.

Thus, we can set the two equations equal to each other and solve for  to find the x-coordinate.

To find the y-coordinate, plug the  back into either of the equations.

Recall how to find the slope of a line:

Since we already have the slope of the line and one coordinate on the line, we will just need to plug in the given answer choices to see which one would give us the correct slope.

Using the coordinate , we would get the following slope:

 must be a point on the given line.

Quadrant III of the rectangular coordinate plane comprises the set of points with its - and -coordinates both negative. Therefore, and must both be negative numbers.

If , then , as the product of a positive number and a negative number, must be a negative number. If , then , as the product of two negative numbers, must be a positive number.

For the same reasons, if then is a negative number, and if , then is a positive number.

For to be in Quadrant III, the only correct choice given would be that  and .

For each point, substitute the first coordinate, or -coordinate, for , and the second coordinate, or -coordinate, for  in the equation. This is done with  to show that this is the correct choice:

By order of operations, work the left multiplication first:

Work the right multiplication next:

Subtract last:

The ordered pair  makes the equality true, so it is the correct choice.

As for the other three, we can work the same steps to demonstrate that each other ordered pair makes the equation incorrect, and, consequently, each is an incorrect choice:

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: 

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